Faber polynomial coefficient estimates for bi-univalent functions defined by subordinations

A function is said to be bi-univalent on the open unit disk D if both the function and its inverse are univalent in D. Not much is known about the behavior of the classes of bi-univalent functions let alone about their coefficients. In this paper we use the Faber polynomial expansions to find coefficient estimates for four well-known classes of bi-univalent functions which are defined by subordinations. Both the coefficient bounds and the techniques presented are new and we hope that this paper will inspire future researchers in applying our approach to other related problems.